There are two main partial results.
First, any two polyominoes of the same area
have a hinged dissection [DDE+03].
A polyomino is a polygon formed by joining unit squares at their
edges; see [Kla97] and Problem 37.
The polyomino result generalizes to hinged dissections
of all edge-to-corresponding-edge gluings
of congruent copies of any polygon.
Second,
any asymmetric polygon has a hinged dissection
to its mirror image [Epp01].
Both of these results interpret the problem as
ignoring possible intersections between
the pieces as they hinge,
following what Frederickson calls the ``wobbly-hinged'' model.
This freedom may not be necessary, although this seems not to
be established in the literature.
Many specific examples of hinged dissections can be found in [Fre02].